Re: Universal grammar

In article <haberg-2610061357150001@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Hans Aberg <haberg@xxxxxxxxxx> wrote:
In article <1161858852.901788.39420@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Franz
Gnaedinger" <frgn@xxxxxxxxxxx> wrote:

I can't help you wit the Goethe idea, not knowing much about the context.

When I was a teenager I got a book on quantum physics.
A footnote said that the basic formula of mathematics,
q equals q, has not yet been investigated. So I began to
ponder this equation. What does it say? It speaks of a thing
or a being named q, and of another thing or being called q,
and it claims that both are identical.

Logic with identity states that it is "full identity",
not that one can replace the other.

Are there any identical
things or beings? really identical ones? Apples vary in form,
size, color and taste.

Again, in logic this is not considered.

..................

The story in working math, is that one pretty much must have a theory with
equality, and therefore there is such a concept also in formal
metamathematics. For example, two sets A, B are equal, written A = B, if
they have the same members, i.e., all x (x in A <=> x in B). Since all
modern math is thought to be expressible using some sort of set theory,
all of it has an equality, at least on the set level.

All forms of set theory assume this for sets, but it is
an axiom, not part of the language. It is part of the
language, the first-order predicate calculus with equality,
that the same operation on equal entities yields equal
results. But one has to be sure it is extensional.

But one does not get to know an answer to thephilosophical question "what
is equality, and why is it there". It is simply so thatexperience has it
that it is useful in theoretical modeling, and therefore it is there from
the practical point of view.

Mathematics does not just mimic nature. It is fully
abstract, with its basic entities purely mental. This
is why it is important to use abstract ideas early, and
with no prior reference to more concrete ones. Going
from more abstract to less abstract is easy; on adds
conditions, so all the previous results hold. Going the
other way is hard; each time conditions are removed, less
of what was true before is true now.

But the point is very much that there is no universal mathematical
grammar. Real working mathematics in reality passes between several formal
theories, each having its own formal language.

This is not the case. The first-order predicate calculus
is the universal language. All theories use this language.
As different axioms are used, different results follow, but
the language used is the same, except for idioms.

So we have reached an agreement. There is no universal
mathematical grammar, ...

That is, from the practical point of view, as we are dealing with humans
that need to write stuffparsableto humans. It is theexperience that in
math, notions and notation must flow together. So this plustradition
mainly dictates thenotational system used.

From the formal point of view, it is believed that all modern math can be
expressed using some sort of set theory (sorry for the repetitions :-)),
so one just has to invent a notation for that, and allmath becomes
expressible. But that is as useful to humans as writingcomputer programs
in machine code only.

...and Einstein's dream of a physics
based solely on mathematical numbers such as 1, 2, pi, e,
won't come true either:

As far as I know, Einstein did not consider that a model of
the real world could be derived from pure reasoning without
taking into account the facts of the world. As with all
post-medieval scientists, he looked for simple models.
However, there is the quote, "Make things as simple as
possible, but no simpler." However, he was not making up
new mathematics to describe physics, but was using what
existest before, and what was known to a fair number.

And was there really a Newton theory that theMoon is a cheese?

...mathematics is based on the formula
a equals a, while language and physics belong to the real
world where Goethe rules: all is equal, all unequal. Note well
that his formula includes the basic equation of mathematics,
and so we can say that mathematics - the logic of building,
constructing, maintaining - is entangled with every part of the
world, while covering only half of the truth. Einstein must have
guessed this; he said mathematics is exact as long as we don't
apply it to problems of the real world, but no longer exact when
applied to a real world problem.

No, it is that the model of the real world is wrong. This
is especially hard when one gets to quantum mechanics,
which Einstein rejected. The observations follow the laws
of probability, which are bad enough; what happens between
observations is not even probability, and only special cases
have been successfully treated.

Most, if not all math, has its origin inmodeling of the real world.
Itevolves with continuos cross-feeding to themodeling of the real world.
For example, Gauss made progress in developing number theory when working
oncomputing planetary orbits - he wasofficially an astronomer, not a
mathematician.

No, he was working on number theory outside his work as
an astronomer. Gauss's thesis was proving the fundamental
theorem of algebra, and this led to number-theoretic
considerations of which regular polynomials could be
constructed by ruler and compass. Having an inventive
mind, he did further work on number theory. Too many
these days are specializing, and losing even in their
specialization by doing so.

In modern days, numbertheory has become veryuseful in
cryptography, for example.

My stand on applied mathematics for decades has not changed.
There is mathematics which has been applied, and mathematics
which has not yet been applied.

Boole made his ideas in the form of logical
modeling, which later became very useful whencomputers as made.

However, he was not thinking of that when he and de Morgan
did their work. Also, Boole wrote on other parts of
mathematics; I found his book on difference equations to
be quite useful.

Cantor
developed his ideas of infinities when working on the convergence of
Fourier series. Newton and Leibnitz, who communicated their ideas of
calculus with each other,got their ideas from Archimedes, I think.And
the so called Pythagorean theorem was known in old Babylon (or
Mesopotamia), for example a 3-4-5 triangle, easy to produce with a piece
of string, was used to measure up land, there and in old Egypt. And
Archimedes realized that the Earth is round, and even measured up its
size. Columbus knew this, of course, but made reasoned thecomputations
were wrong error, because he then upper his chances to get money for the
trip he eventually made.

The Babylonians used the 3-4-5 right triangle. They also
knew of other integer Pythagorean triples, but it is not
clear that they knew these also formed right triangles.
It seems it was an arithmetical exercise.

It was Eratosthenes, long before Archimedes, who measured
the size of the Earth.

This can also be seen in
language. I remember an article in The American Scientist
on the ambiguities of English words such as same, equal,
identical. These words are fuzzy, depending on the context.
Identical in a "normal" sense means very much the same,
equal in all important features; in mathematics the same word
has a different meaning, really really identical, in every aspect.
I don't see any way to get rid of such ambiguities, ...

In math, there are no such ambiguities. For example, "identical" might
mean equal as sets. The one defines an equivalence relation, which defines
the equalityused in practice.

..and assume
they belong intrinsically to language that works on many levels
at the same time. The wonder of language is that we can say
so much about the world with a limited set of some 26 letters,
a b c d e f g h i j k l m n o p q r s t u v w x y z, 10 digits, 0
1 2 3 4 5 6 7 8 9, and some extra signs.

Unicode has some 100000 plus characters in it, I think. And, in the other
direction, any positive integer can be expressed using "unary" notation
1...1. So you just need a symbol '1' and a symbol like ' ' that separates
the numbers.

This wonder is only
possible by means of ambiguity - a negative term, should
I better say polyvalency?

I think ambiguity only arises because humans are used to, andeasily can
cope with such. They are not as such needed for the development of a
formal theory.

--
Hans Aberg


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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